Concept Architecture
Concept
Theoretically, the Tarone-Ware Test is a weighted non-parametric hypothesis test used to compare survival distributions between two or more groups. It is a member of the weighted log-rank family of tests and differs from the standard log-rank test by assigning greater weight to failures occurring earlier in follow-up while still considering events throughout the study period. The method exists because treatment effects may not be constant over time, and increased sensitivity to early differences may be desirable in certain clinical settings.
Mathematically, the Tarone-Ware test modifies the log-rank test statistic by applying weights proportional to the square root of the number of individuals at risk at each event time. The weighted observed-minus-expected event differences are summed across all event times and standardised by their variance to produce a chi-square test statistic. Under the null hypothesis of equal survival functions, the statistic follows an approximate chi-square distribution.
In practice, the Tarone-Ware test is applied when survival curves separate predominantly during the early or middle stages of follow-up. It is frequently used alongside the log-rank and Wilcoxon tests as a sensitivity analysis in clinical trials and observational studies that inform survival modelling and health economic evaluations.
Purpose
Used to compare survival distributions while placing greater emphasis on earlier event times than the standard log-rank test, improving sensitivity when treatment effects occur predominantly during the initial phases of follow-up.
Mathematical Formulae
Primary Formula
?� = (?w?(O? ? E?))� / ?w?�V?
where:
- w? = �n?
- n? = number at risk immediately before event time i
- O? = observed events
- E? = expected events
- V? = variance at event time i
Supporting Formulae
Weight function:
w? = �n?
Standardised statistic:
Z = ?w?(O? ? E?) / �?w?�V?
Chi-square statistic:
?� = Z�
Related Mathematical Methods
- Log-Rank Test
- Wilcoxon (Breslow?Gehan) Test
- Fleming?Harrington Tests
- Kaplan?Meier Estimation
- Survival Analysis
- Cox Proportional Hazards Model
Example
A clinical trial compares two cancer therapies over five years. The largest differences in survival occur during the first 18 months.
At successive event times, the weighted observed-minus-expected event differences are combined using Tarone-Ware weights based on the square root of the number at risk.
The final analysis produces:
?� = 5.92
p = 0.015
The result indicates a statistically significant difference in survival, with the Tarone-Ware test providing greater sensitivity to the early separation of the survival curves than the standard log-rank test.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SQRT | =SQRT(B2) | Calculate Tarone-Ware event weights from the number at risk |
| SUMPRODUCT | =SUMPRODUCT(Weights,ObservedMinusExpected) | Calculate the weighted numerator of the test statistic |
| SUMPRODUCT | =SUMPRODUCT((Weights^2),Variance) | Calculate the weighted variance |
| CHISQ.DIST.RT | =CHISQ.DIST.RT(TestStatistic,1) | Calculate the p-value |
| IF | =IF(P2<0.05,"Significant","Not Significant") | Interpret the statistical result |
VBA (Optional)
A VBA procedure can automate weighted survival comparisons by calculating Tarone-Ware statistics and generating comparative reports for multiple treatment groups.
Sources
- Tarone RE, Ware J. On Distribution-Free Tests for Equality of Survival Distributions. Biometrika. 1977.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Collett D. Modelling Survival Data in Medical Research.
- Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 15: Cost-effectiveness modelling using patient-level simulation — Davis, Stevenson, Tappenden & Wailoo, TSD 15 ed., 2014 (NICE Decision Support Unit (University of Sheffield))
Guidance on individual patient-level (microsimulation) cost-effectiveness modelling — when to use it in preference to cohort models, how to structure it, and how to handle the associated computational and uncertainty challenges.
Frequently Asked Questions (6)
What is the Tarone-Ware test?
A test comparing survival distributions using a weighting scheme between the log-rank and Peto tests, a compromise depending on when treatment differences emerge.
Source: Tarone & Ware 1977
Where does the Tarone-Ware test sit among weighted survival tests?
The weighted family of survival comparisons differs in how much emphasis each places on early versus late event times. The log-rank test weights all times equally, the Wilcoxon-type test weights early events most heavily, and the Tarone-Ware test sits between them, using a weight based on the square root of the number at risk. This middle position makes it a compromise, less extreme than either in where it concentrates its sensitivity. The choice among them depends on when a difference is expected to emerge. Klein and Moeschberger (2003) describe these tests.
Source: Klein & Moeschberger 2003
How does the Tarone-Ware test weight events?
The Tarone-Ware test weights events at each time by the square root of the number at risk, which gives more weight to earlier events, when more are at risk, than to later ones, but less strongly than the Wilcoxon test, which weights by the number at risk itself. This places its weighting between the log-rank test, which weights all events equally, and the Wilcoxon test. The square-root weighting is the compromise that characterises the Tarone-Ware test.
Source: Tarone & Ware 1977
How does the Tarone-Ware test compare with the log-rank and Wilcoxon tests?
The Tarone-Ware, log-rank, and Wilcoxon tests belong to the family of weighted rank tests but differ in weighting: the log-rank test weights all events equally and is most powerful under proportional hazards; the Wilcoxon test weights by the number at risk, emphasising early events; and the Tarone-Ware test uses the square root of the number at risk, an intermediate weighting. So the Tarone-Ware test is a compromise, more sensitive to early differences than the log-rank test but less so than the Wilcoxon test.
Source: Fleming & Harrington 1991
When is the Tarone-Ware test used?
The Tarone-Ware test is used to compare survival between groups when a weighting between the equal emphasis of the log-rank test and the strong early emphasis of the Wilcoxon test is wanted, offering a balanced sensitivity to differences arising at different times. It is chosen where the timing of any survival difference is uncertain and a compromise weighting is preferred, or as one of several weighted tests considered. As with other weighted tests, the choice should ideally be specified in advance.
Source: Tarone & Ware 1977
What determines the choice of the Tarone-Ware test among weighted survival tests?
The choice among weighted survival tests, such as the log-rank, Wilcoxon, and Tarone-Ware, depends on when survival differences between groups are expected to emerge: the log-rank test suits proportional hazards and differences spread over time, the Wilcoxon test suits early differences, and the Tarone-Ware test is a compromise. The weighting should be chosen in advance based on the anticipated pattern, since selecting it after seeing the data to maximise significance would invalidate the test. The tests differ in the pattern of difference they best detect.
Source: Tarone & Ware 1977
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British health economist
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Verification date: 24 Oct 2025
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